Riemann k-surfaces in multivariable control systems

Riemann surfaces were introduced in control theory to solve ambiguous situations in multivariable control systems. Square systems with m inputs and m outputs have Riemann surfaces in m sheets or copies of the complex plane. We show that it is possible to represent all sheets of a Riemann surface in a unique graph, and we show how to plot this graph for the Riemann k-surfaces of a square multivariable system. Different representations appear as some variables´ combinations, as the studied space is 4-dimensional: system poles s frequency have complex values as function of the complex values imposed on the gain variable k. It is shown that these k-surfaces include the root locus and complementary root locus branches, and graphic forms are shown for the k-surfaces, which include generalization of gain plots and other representations for root locus and complementary root locus. 3-dimensional representations for Riemann k-surfaces are also shown